The lattice-tree and lattice-animal finite-size scaling profile conjecture

Lattice trees and lattice animals are finite connected subgraphs and acyclic connected subgraphs, respectively, of the lattice. On the torus $\mathbb{T}_r^d$ with volume $V=r^d$, let $G_{r,z}(x)$ denote the two-point function, let $G_{z_c}(x)$ denote its infinite-volume critical counterpart, let $\chi_r(z)$ denote the susceptibility, and let $I_0$ be the rescaled profile introduced in the paper. For $d>8$, there are constants $a_d<0$ and $b_d>0$, different for trees and animals, such that, as $V=r^d\to\infty$,

$$G_{r,z_c+sV^{-1/2}}(x)-G_{z_c}(x)\sim b_dV^{-3/4}I_0(a_ds),$$ $$\chi_r(z_c+sV^{-1/2})\sim b_dV^{1/4}I_0(a_ds).$$

Lattice-tree and lattice-animal profile conjecture. The same critical finite-size scaling profile $I_0$ occurs for both lattice trees and lattice animals on $\mathbb{T}_r^d$ above the upper critical dimension. This is motivated by the corresponding profile for trees and connected subgraphs on the complete graph and by the expected high-dimensional behaviour of lattice trees and animals; the claim remains conjectural in the stated setting.